Hesse normal form
The HNF, Hesse normal form or hess ash normal form is in mathematics, a special form of a linear equation or plane equation. In between hesse normal form a Euclidean straight line in the plane or a plane in Euclidean space by the distance from the origin and a normalized and oriented normal vector is shown. In between hesse normal form thus is a special implicit representation of the straight line or plane.
The hess ash Normal Form allows an efficient calculation of the distance of any point on the line or plane. It is named after the German mathematician Otto Hesse.
- 2.1 Presentation
- 2.2 Example
- 2.3 Calculation
- 2.4 distance
- 3.1 hyperplanes
- 3.2 Curves and Surfaces
Hessian normal form of a linear equation
Representation
In between hesse normal form a straight line in the Euclidean plane by a normalized and oriented normal vector of the line and its distance from the origin is described. A straight line then consists of those points in the plane, whose position vectors of the equation
. meet Here denotes the scalar product of two vectors. The normal vector is a vector that forms a right angle with the straight line. He must have the length and show the coordinate origin in the direction of the straight line, so it must be true.
Accordingly, the points are implicitly defined by the straight line in the HNF that the dot product of the position vector of a line point and the normal vector of the straight line is equal to the distance of the straight line from the origin. A point whose position vector does not satisfy the equation, is located on the side of the straight line in which the normal vector indicates otherwise, and on the other side. The coordinate origin is always located on the negative side of the line, if it is not a line through the origin.
Example
For example, a normalized normal vector of a given line and the distance of the line from the origin, we obtain the linear equation
Each choice of which satisfies this equation, for example, or corresponds to, then a line point.
Calculation
From the normal form of a linear equation with support vector and normal vector to a normalized and oriented normal vector of the straight line can be carried
Determine. The distance of the line from the origin can then
Be determined. This distance corresponds exactly to the length of the orthogonal projection of the vector onto the line through the origin with direction vector.
From the other forms of linear equations, the coordinate form, intercept form, the parameter form and the two-point form, the associated normal form of the straight line is first determined ( see calculation of the normal form) and from this the hess ash normal form.
Distance
With the help of hesse between normal form of the distance of any point can be in the plane of a straight line easily be calculated that the position vector of the point is used in the linear equation:
This distance has a sign: for the point lies on the side of the straight line in which the normal vector indicates otherwise, on the other hand.
Hessian normal form of a plane equation
Representation
Analog describes a plane in three-dimensional space in between hesse normal form by a normalized and oriented normal vector of the plane and its distance from the origin. A level then consists of those points in space whose position vectors, the equation
. meet The normal vector here is a vector which is perpendicular to the plane. The normal vector must again have the length and show the origin of coordinates in the plane, so it must be true.
Thus the points of the plane are implicitly defined by In the hesse between normal form that the scalar product of the position vector of a plane point and the normal vector of the plane is equal to the distance of the plane from the origin. Again there is a point whose location vector satisfies the equation on the plane. Holds, then the point is on the side of the plane in which the normal vector indicates otherwise, on the other hand. The coordinate origin is always located on the negative side of the plane, if it is not original plane.
Example
For example, a normalized normal vector of a given level and the distance of the plane from the origin, we obtain the plane equation
Each choice of which satisfies this equation, for example, or corresponds to, then a point of the plane.
Calculation
From the normal form of a plane equation with support vector and normal vector to a normalized and oriented normal vector of the plane as in the two-dimensional case can be carried
Determine. The distance of the plane from the origin can then
Be determined. This distance, in turn, corresponds to the length of the orthogonal projection of the vector onto the line through the origin with direction vector.
From the other forms of plane equations, the coordinate form, intercept form, the parameter form and the three -point form, the associated normal form of the plane is first determined ( see calculation of the normal form) and from this the hess ash normal form.
Distance
Using the HNF the distance of any point in space by a plane in turn can be calculated, characterized in that the position vector of the point is inserted into the plane equation:
This distance is again signed: for the point lies on the side of the plane in which the normal vector indicates otherwise, on the other hand.
Generalizations
Hyperplanes
In general, a hyperplane is described in the - dimensional Euclidean space by the hess ash normal form. In - dimensional Euclidean space is a hyperplane in accordance of those points whose position vectors of the equation
. meet It is merely expected -component instead of two - or three-component vectors. A hyperplane divides the -dimensional space into two parts, which are called half-spaces. A point whose position vector satisfies the equation, is located exactly on the hyperplane. Holds, then the point lies in the one half-space, in which the normal vector points, otherwise in the other.
Curves and Surfaces
The normal form is also available for plane curves. It is an implicit representation
A curve with the property. For example,
The normal form of the circle. The function describes the oriented distance from a point to the curve and is called the distance function. For surfaces, there is the normal form. Normal forms for curves and surfaces have both theoretical significance (analogous to the Bogenlängenparametrisierung of curves) as well as practical in geometric modeling.